How do you solve 8x^3-24x^2+12x-4=0?

first step: divide by 4 to make the numbers more manageable:

2x^3 - 6x^2 + 3x - 1 = 0

Now you know that any rational roots will be ±1 or ±1/2

So, try them out. Turns out that none of them is a root.

You can use Descarte's Rule of Signs to show that there are no negative roots, and at most three positive roots (duh).

So, now you have to fall back on the general solution of cubics (very difficult and messy) or iterative methods to get an approximate solution to any desired accuracy.

To solve the equation 8x^3 - 24x^2 + 12x - 4 = 0, you can use a method called the Rational Root Theorem, followed by synthetic division or long division. Here's how it works:

Step 1: Apply the Rational Root Theorem
The Rational Root Theorem states that if a rational number p/q is a root of the polynomial equation, where p is a factor of the constant term (-4 in this case) and q is a factor of the leading coefficient (8 in this case), then p/q will be a root of the equation.

So, to find the possible rational roots, you need to list all the factors of -4 and 8:
Factors of -4: ±1, ±2, ±4
Factors of 8: ±1, ±2, ±4, ±8

Step 2: Test the Possible Rational Roots
Once you have the list of possible rational roots, you can test each one by substituting them into the equation and check if they satisfy it.

Synthetic Division:
Let's assume you've found a rational root, let's say x = a, where a is a rational number that you choose from the list of possible roots obtained in Step 1.

To do synthetic division:
1. Write down the coefficients of the polynomial equation: 8, -24, 12, -4, and the assumed root, a.
2. Perform synthetic division to check if the assumed root is a factor of the polynomial equation. The last value of the synthetic division is the remainder.

If the remainder is zero (0), then a is a root of the equation and (x-a) is a factor of the polynomial equation. If the remainder is non-zero, repeat the synthetic division with a different root until you find a root that gives a remainder of zero.

Step 3: Solve the Reduced Equation
If you have found a root using synthetic division, you can divide the original polynomial by (x - a) to obtain a reduced equation, which is a quadratic equation (since the original equation is a cubic). Then, solve the quadratic equation using factoring, completing the square, or the quadratic formula.

By following these steps, you should be able to find the solutions to the equation 8x^3 - 24x^2 + 12x - 4 = 0.